A very basic doubt - What is the value of a callable bond when the bond is ITM

For puts I understand it. I am the holder and if the option is ITM I gain from it, but when a callable is ITM why is it valued higher? Do I get compensated more by the issuer?

Callable bond should have a higher yield/lower price than non-option bond.
Issuer has benefit / investor has extra risk (ltd no upside at low yields) and thus demands a higher price. The more “ITM” the call is the further the callable bond will be from the normal bond price.

Ohh gotcha. Just so that I understood this well, irrespective of the fact if a callable bond is ITM or ATM, its value cannot be more than a straight bond or else it will be called. As for put if the bond is OTM, the value should be appx the same as a straight bond but if it is ITM the value will be higher?

It is not the value of the straight option that sets the cap it is the call price (although in realty it is a little more complex. Callable bonds are usually only callable on certain dates not every day, as the bond may not be callable for a number of years and interest rates low so investors will be willing to pay more than call price just like they will pay more than par)

The call/put price is not a strict cutoff point. The diagram above does not take a abrupt turn at the call price. The market discounts the effect of the call/put. The line is more like a curve.

We would a expect a callable to always sell below teh staright bond. Along way from the call price this difference as we get same but as we get closer and the difference will get bigger. Note the graph above.

We expect a putable bond to always above the straight bond. By a small amount when we are along way from teh put price and the difference will get bigger as we get closer to the put price.

Got it. Thank you!